Changes

4. A premise that is better known than the conclusion has the ''[[genus]]'' (or relevant class to which the subject belongs) of the conclusion's subject for its own subject or appropriately in its predicate, and to be prior to the conclusion means the [[genus]] in the premise is ''not somehow determined'' by the subject of the conclusion that belongs to it.

4. A premise that is better known than the conclusion has the ''[[genus]]'' (or relevant class to which the subject belongs) of the conclusion's subject for its own subject or appropriately in its predicate, and to be prior to the conclusion means the [[genus]] in the premise is ''not somehow determined'' by the subject of the conclusion that belongs to it.

5. Since premises are to be related to their conclusion through cause and effect, they must be as ''necessarily'' true as the desired necessary truth of the conclusion would demand: either ''rigorously proven'' themselves or ''self-evident'', without the truth or its necessity narrowing as it goes down the syllogistic chain to the point where it could no longer apply to the conclusion.

5. Since premises are to be related to their conclusion through cause and effect, they must be as ''necessarily'' true as the desired necessary truth of the conclusion would demand: either ''rigorously proven'' themselves, ''self-evident'' or what is asserted to be ''common sense'' (by definition antecedent to philosophical distinctions), without the truth or its necessity narrowing as it goes down the syllogistic chain to the point where it could no longer apply to the conclusion.

However, a state of "conditional necessity" may exist in the premises, so long as the condition is not determined by the subject of the conclusion as per requirement 4, but is rather an "assumption for a conditional proof", an assumption which would appear somewhere in the validly co-ordinated and soundly applied series of premises.

However, a state of "conditional necessity" may exist in the premises, so long as the condition is not determined by the subject of the conclusion as per requirement 4, but is rather an "assumption for a conditional proof", an assumption which would appear somewhere in the validly co-ordinated and soundly applied series of premises.

As an example, a conditional proof or demonstration would fail its "better-known" soundness criterion if one assumed as a condition "eating nuts make me allergic" and yet conclude in a simple way, "Walnuts are nuts with hard shells; I can eat walnuts and not get allergic."  The would-be conclusion would need to attempt to redetermine the thought-to-be "better-known" genus of "nuts" in the condition "eating nuts make me allergic" with the implied, qualified subject of the dubious conclusion, "soft-shelled nuts", so that, until that happens, the premise containing the genus "nuts" would no longer be better-known than the conclusion due to that premise having notions about the species "hard-shelled nuts" that contradict information alleged in the conclusion.

As an example, a conditional proof or demonstration would fail its "better-known" soundness criterion if one assumed as a condition "eating nuts make me allergic" and yet conclude in a simple way, "Walnuts are nuts with hard shells; I can eat walnuts and not get allergic."  The would-be conclusion would need to attempt to redetermine the thought-to-be "better-known" genus of "nuts" in the condition "eating nuts make me allergic" with the implied, qualified subject of the dubious conclusion, "soft-shelled nuts" or else itself be corrected.

 

So that, until that happens, the conditional premise, whatever use it eventually has, containing the genus "nuts" would no longer be better-known than the conclusion due to that conditional premise having (implied) notions about the species "hard-shelled nuts" that contradict information, or perhaps don't allow for an exception, alleged in the conclusion.

==Mathematical proof==

==Mathematical proof==